38 Prof. F. Slate on a 



Accordingly an "apparent" force (T a ') estimated in that 

 (U) is given by 



T / dv . .dm d, rx T dm f ox 



T a =m-£ +(«.-«) - dt = -^)=T -u^ . . (12) 



Add this explicit agreement : Effective inertia is inferred 

 from physical relations in (F), and passes unchanged through 

 such purely kinematicai revisions. Direct combination of 

 equations (4, 12} under their condition of equal accelerations 

 shows 



T„'=(l-'-^)T„ (13) 



The "reduction factor" (l — uv /c 2 ) connects the forces 

 manifest in (U, F) through using (m) with the respective 

 kinematicai data. Or equation (12) is evidently applicable 

 to compare forces in (F), for the same inertia and acceleration y 

 at differing speeds. 



In particular at (w==v ), 



T a '=m-~ [u = v ] (14) 



This introduces the familiar "rest-system" of relativity* 

 And since we may write 



T/=[(i-^) mo y W ]§ sm 



, dv 

 dt 



under the suggestion of rigid dynamics, it would follow that 



?=^ T -=te=^ w M-*-5(- • •• (15) 



Identification of " FresnePs coefficient " (k) in this second 

 member fits the idea of its connexion with "inertia-drag'"' 

 in (U) *. 



Einstein's "Addition theorem for velocities" presents the 

 same reduction factor as equation (13). But there it would 

 reduce the velocity of (m) relative to (O') the origin of (U) 

 to dependence upon specifying frames. Newton makes that 

 a constant difference of any pair of corresponding values. 

 The intrinsic meaning of the theorem must go deeper, since 

 its characteristic factor offers itself unconstrainedly in a 

 dynamic relation of the two frames. This must quicken the 

 surmise that what underlies the " Lorentz transformation" 



* Cf. Silberstein, pp. 172-3. 



